Subgroups — C₁₂₇×#21821 #44585494

Order 32512.

1014 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#432×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#431×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1985	C₂	1	4	yes	maximal, complemented
5	C₁₂₇×#1984	C₂	1	4	yes	maximal, complemented
6	C₁₂₇×#1818	C₂	1	4	yes	maximal
7	#533×C₁₂₇	—	2	4	no	—
8	#533×C₁₂₇	—	2	4	no	—
9	C₂×C₄⋊Q₈×C₁₂₇	C₄	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₁₂₇	C₄	1	1	yes	characteristic
11	C₂×#125×C₁₂₇	C₄	1	2	yes	—
12	C₂×C₂²⋊Q₈×C₁₂₇	C₄	1	2	yes	—
13	C₂×C₂²⋊Q₈×C₁₂₇	C₄	1	2	yes	—
14	C₂×C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
16	C₂×#117×C₁₂₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
18	#432×C₁₂₇	—	2	4	no	—
19	#431×C₁₂₇	—	2	4	no	—
20	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
24	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
26	#336×C₁₂₇	C₂²	1	16	yes	—
27	C₂²×Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
28	C₂²×Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₂₇	C₂³	1	4	yes	—
33	C₂²×C₈×C₁₂₇	C₂³	1	4	yes	—
34	C₄⋊Q₈×C₁₂₇	—	2	4	no	—
35	C₄⋊Q₈×C₁₂₇	—	2	4	no	—
36	#125×C₁₂₇	—	2	8	no	—
37	C₂²⋊Q₈×C₁₂₇	—	2	8	no	—
38	C₂²⋊Q₈×C₁₂₇	—	2	8	no	—
39	C₄×Q₈×C₁₂₇	—	2	4	no	—
40	C₄×Q₈×C₁₂₇	—	2	4	no	—
41	C₄×Q₈×C₁₂₇	—	2	4	no	—
42	C₄×Q₈×C₁₂₇	—	2	4	no	—
43	#117×C₁₂₇	—	2	4	no	—
44	#117×C₁₂₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
49	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₁₂₇	D₄	1	4	yes	—
55	C₂×C₄²×C₁₂₇	D₄	1	4	yes	—
56	C₂×C₄²×C₁₂₇	C₂³	1	4	yes	—
57	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₁₂₇	—	2	4	no	—
60	C₄⋊C₈×C₁₂₇	—	2	4	no	—
61	C₂²⋊C₈×C₁₂₇	—	2	4	no	—
62	C₂²⋊C₈×C₁₂₇	—	2	4	no	—
63	#97×C₁₂₇	—	2	4	no	—
64	C₄×C₈×C₁₂₇	—	2	4	no	—
65	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×Q₈×C₁₂₇	—	2	4	no	—
67	C₂×Q₈×C₁₂₇	—	2	4	no	—
68	C₂×Q₈×C₁₂₇	—	2	8	no	—
69	C₂×Q₈×C₁₂₇	—	4	4	no	—
70	C₂²×C₄×C₁₂₇	—	2	4	no	—
71	C₂²×C₄×C₁₂₇	—	2	4	no	—
72	C₂²×C₄×C₁₂₇	—	4	2	no	—
73	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
74	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
75	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₁₂₇	—	2	4	no	—
86	C₂×C₈×C₁₂₇	—	2	4	no	—
87	C₂×C₈×C₁₂₇	—	2	8	no	—
88	C₂×C₈×C₁₂₇	—	2	8	no	—
89	C₄⋊C₄×C₁₂₇	—	2	4	no	—
90	C₄⋊C₄×C₁₂₇	—	2	4	no	—
91	C₄⋊C₄×C₁₂₇	—	2	8	no	—
92	C₄⋊C₄×C₁₂₇	—	2	8	no	—
93	C₄⋊C₄×C₁₂₇	—	2	4	no	—
94	C₄⋊C₄×C₁₂₇	—	2	4	no	—
95	C₂²⋊C₄×C₁₂₇	—	2	8	no	—
96	C₄²×C₁₂₇	—	2	4	no	—
97	C₄²×C₁₂₇	—	2	4	no	—
98	C₄²×C₁₂₇	C₂²⋊C₄	1	4	yes	—
99	C₄²×C₁₂₇	C₂²⋊C₄	1	4	yes	—
100	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
101	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
102	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
103	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
104	C₂³×C₁₂₇	—	2	2	no	—
105	C₂³×C₁₂₇	—	2	2	no	—
106	C₂³×C₁₂₇	—	2	2	no	—
107	C₂³×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
109	Q₈×C₁₂₇	—	4	8	no	—
110	Q₈×C₁₂₇	—	4	8	no	—
111	C₂×C₄×C₁₂₇	—	2	2	no	—
112	C₂×C₄×C₁₂₇	—	2	2	no	—
113	C₂×C₄×C₁₂₇	—	2	2	no	—
114	C₂×C₄×C₁₂₇	—	2	2	no	—
115	C₂×C₄×C₁₂₇	—	2	4	no	—
116	C₂×C₄×C₁₂₇	—	2	4	no	—
117	C₂×C₄×C₁₂₇	—	2	8	no	—
118	C₂×C₄×C₁₂₇	—	2	4	no	—
119	C₂×C₄×C₁₂₇	—	2	4	no	—
120	C₂×C₄×C₁₂₇	—	2	8	no	—
121	C₂×C₄×C₁₂₇	—	2	8	no	—
122	C₂×C₄×C₁₂₇	—	4	4	no	—
123	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
124	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
125	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
126	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
127	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂×C₄×C₁₂₇	#131	1	4	yes	—
129	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₂₇	2₋¹⁺⁴	1	4	yes	—
132	C₈×C₁₂₇	—	4	4	no	—
133	C₈×C₁₂₇	—	4	4	no	—
134	C₂²×C₁₂₇	—	2	2	no	—
135	C₂²×C₁₂₇	—	2	2	no	—
136	C₂²×C₁₂₇	—	2	2	no	—
137	C₂²×C₁₂₇	—	4	4	no	—
138	C₂²×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₁₂₇	#519	1	2	yes	—
140	C₂²×C₁₂₇	C₂×#131	1	2	yes	—
141	C₂²×C₁₂₇	#568	1	2	yes	—
142	C₄×C₁₂₇	—	2	2	no	—
143	C₄×C₁₂₇	—	2	2	no	—
144	C₄×C₁₂₇	—	2	2	no	—
145	C₄×C₁₂₇	—	2	2	no	—
146	C₄×C₁₂₇	—	4	4	no	—
147	C₄×C₁₂₇	—	4	4	no	—
148	C₄×C₁₂₇	#408	1	4	yes	—
149	C₄×C₁₂₇	#536	1	2	yes	—
150	C₄×C₁₂₇	#536	1	2	yes	—
151	#21821	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂×C₁₂₇	—	4	2	no	—
153	C₂×C₁₂₇	#2310	1	1	yes	characteristic
154	C₂×C₁₂₇	C₂×#408	1	1	yes	characteristic
155	C₂×C₁₂₇	#2879	1	1	yes	characteristic
156	C₂×C₁₂₇	#2985	1	4	yes	—
157	C₂×#533	C₂×C₁₂₇	1	1	yes	characteristic
158	C₂×#432	C₂×C₁₂₇	1	1	yes	characteristic
159	C₂×#431	C₂×C₁₂₇	1	1	yes	characteristic
160	#1985	C₂×C₁₂₇	1	4	yes	complemented
161	#1984	C₂×C₁₂₇	1	4	yes	complemented
162	#1818	C₂×C₁₂₇	1	4	yes	—
163	C₁₂₇	#21821	1	1	yes	characteristic, direct product factor, Sylow
164	#533	—	2	4	no	—
165	#533	—	2	4	no	—
166	C₂×C₄⋊Q₈	C₄×C₁₂₇	1	1	yes	characteristic
167	C₂×C₄⋊Q₈	C₄×C₁₂₇	1	1	yes	characteristic
168	C₂×#125	C₄×C₁₂₇	1	2	yes	—
169	C₂×C₂²⋊Q₈	C₄×C₁₂₇	1	2	yes	—
170	C₂×C₂²⋊Q₈	C₄×C₁₂₇	1	2	yes	—
171	C₂×C₄×Q₈	C₂²×C₁₂₇	1	2	yes	—
172	C₂×C₄×Q₈	C₂²×C₁₂₇	1	2	yes	—
173	C₂×#117	C₂²×C₁₂₇	1	2	yes	—
174	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
175	#432	—	2	4	no	—
176	#431	—	2	4	no	—
177	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
178	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
179	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
180	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
181	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
182	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
183	#336	C₂²×C₁₂₇	1	16	yes	—
184	C₂²×Q₈	C₂×C₄×C₁₂₇	1	2	yes	—
185	C₂²×Q₈	C₂×C₄×C₁₂₇	1	2	yes	—
186	C₂³×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
187	C₂³×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
188	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₁₂₇	1	4	yes	—
190	C₂²×C₈	C₂³×C₁₂₇	1	4	yes	—
191	C₄⋊Q₈	—	2	4	no	—
192	C₄⋊Q₈	—	2	4	no	—
193	#125	—	2	8	no	—
194	C₂²⋊Q₈	—	2	8	no	—
195	C₂²⋊Q₈	—	2	8	no	—
196	C₄×Q₈	—	2	4	no	—
197	C₄×Q₈	—	2	4	no	—
198	C₄×Q₈	—	2	4	no	—
199	C₄×Q₈	—	2	4	no	—
200	#117	—	2	4	no	—
201	#117	—	2	4	no	—
202	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
203	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
204	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
205	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
206	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
207	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
208	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
209	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
210	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
211	C₂×C₄²	D₄×C₁₂₇	1	4	yes	—
212	C₂×C₄²	D₄×C₁₂₇	1	4	yes	—
213	C₂×C₄²	C₂³×C₁₂₇	1	4	yes	—
214	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
216	C₄⋊C₈	—	2	4	no	—
217	C₄⋊C₈	—	2	4	no	—
218	C₂²⋊C₈	—	2	4	no	—
219	C₂²⋊C₈	—	2	4	no	—
220	#97	—	2	4	no	—
221	C₄×C₈	—	2	4	no	—
222	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
223	C₂×Q₈	—	2	4	no	—
224	C₂×Q₈	—	2	4	no	—
225	C₂×Q₈	—	2	8	no	—
226	C₂×Q₈	—	4	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
231	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
232	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
233	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
234	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
235	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
236	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
237	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
238	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
241	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	8	no	—
245	C₂×C₈	—	2	8	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	8	no	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₂²⋊C₄	—	2	8	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	C₂²⋊C₄×C₁₂₇	1	4	yes	—
256	C₄²	C₂²⋊C₄×C₁₂₇	1	4	yes	—
257	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
258	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
259	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
260	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	2	no	—
264	C₂³	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
265	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
266	Q₈	—	4	8	no	—
267	Q₈	—	4	8	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
281	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
282	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
283	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
284	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
285	C₂×C₄	#131×C₁₂₇	1	4	yes	—
286	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
288	C₂×C₄	2₋¹⁺⁴×C₁₂₇	1	4	yes	—
289	C₈	—	4	4	no	—
290	C₈	—	4	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	4	4	no	—
295	C₂²	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
296	C₂²	#519×C₁₂₇	1	2	yes	—
297	C₂²	C₂×#131×C₁₂₇	1	2	yes	—
298	C₂²	#568×C₁₂₇	1	2	yes	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	4	4	no	—
304	C₄	—	4	4	no	—
305	C₄	#408×C₁₂₇	1	4	yes	—
306	C₄	#536×C₁₂₇	1	2	yes	—
307	C₄	#536×C₁₂₇	1	2	yes	—
308	C₂	—	4	2	no	—
309	C₂	C₁₂₇×#2310	1	1	yes	characteristic
310	C₂	C₂×#408×C₁₂₇	1	1	yes	characteristic
311	C₂	C₁₂₇×#2879	1	1	yes	characteristic
312	C₂	C₁₂₇×#2985	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂³×C₁₂₇	C₂³×C₄
prime radical	C₂⁴×C₁₂₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₂₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. C₁₂₇×#1985	C₂	semidirect
5. C₁₂₇×#1984	C₂	semidirect
151. #21821	C₁₂₇	direct (N × H)
160. #1985	C₂×C₁₂₇	semidirect
161. #1984	C₂×C₁₂₇	semidirect
163. C₁₂₇	#21821	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#533×C₁₂₇	2
C₂×#432×C₁₂₇	2
C₂×#431×C₁₂₇	2
C₁₂₇×#1985	2
C₁₂₇×#1984	2
C₁₂₇×#1818	2
#21821	127

Subgroups — C127×#21821 #44585494

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21821 #44585494